Dan Henderson, Millington, MD, DanielHenderson@cpm.org

I’ve been thinking about how to normalize students changing their minds in math class. The way I see it, you have to change your mind to learn. That’s, like, what learning is. I mean, I’ve heard Daniel Willingham misquoted as saying that learning is the residue of thought, but that’s memory. Memory is the residue of thought. Learning is letting your ideas and conceptions evolve. So I’ve been thinking about how to normalize the evolution of ideas in math class.

How can I consistently cultivate kaizen (constant improvement) in my students? I have made a few steps in that direction, but no game-changers. I’ve shown videos praising the elasticity of brains; displayed slogans like “Every expert was once a beginner” and “Mistakes are expected, inspected, and respected;” and preached the power of “yet” on several occasions. But how can I move beyond slogans and one-off interventions toward perpetual progress?

I’m always tempted to declare “Everything is hopeless” at this time of year, but the truth is, it’s not. Though it is dark outside, our future is bright. We can do several things to normalize changing minds in math class. Here are a few within our power:

  • Present specialized problems. Take the first few minutes of class to invite students to change their responses several times in a row with a slow reveal, encourage students to generate insights through trial and error in a dakabibi*, or work out a definition by considering examples and non-examples.
  • Work on erasable surfaces. Say, “Consider changing your plan” by having students work on whiteboards, chalkboards, windows, laminated paper, sheet protectors, or repurposed shower curtains.
  • Invite metacognition. Prompt students to revisit ideas and conceptions with journal prompts, self-assessments, and rubrics that provide feedback instead of a grade. Provide plenty of white space in notes for students to annotate solutions and record their own takeaways.
  • Implement routines. Structure as many interactions as you can around iteration and revision. Call out Rough Draft Talk, use language routines like Stronger and Clearer each time, and incorporate talk moves to help students build on and modify ideas even if they don’t have their own starting point.
  • Use teacher moves. Reframe changing your work as a positive move forward. Ask students to estimate first or give wrong answers only. Bring students into an evolving problem landscape by launching tasks orally and introducing constraints as they come up naturally. Model changing your mind by making declarative statements and having students talk you out of them. Lower the stakes by having students discuss with partners.
  • Position all contributions as helpful. Amplify contributions of all types, including those made by students’ gestures, pictures, and phrases borrowed from other languages. Make visible how all students’ representations connect. We need to go beyond labeling ideas as “brilliant” and “wrong” and recognize that student responses most often represent their best understanding given their current model and information. It is not a misconception, but rather their at-the-moment conception based on their knowledge. Connect misdirected, circuitous, and partial solutions to the learning goal during lesson consolidation. Encourage students to learn from each other using Swapmeet and Stop and Scan (or I Spy) strategies. Say things like, “Sasha, can you share what you said to your team with the class? I thought that was really helpful.”
  • Control our own reactions. Focus students on the process, not the answer. Students need affirmation, not answers; they are often more in need of encouragement than help. Listen to what students say and ask them about it. Respond to questions with questions that push students to think deeper about the problem. Don’t be afraid to leave some questions unanswered.

Constant improvement in nature is commonplace, conventional, and even necessary. It is natural that things and ideas evolve. We should anticipate change and growth every day. My quest to renormalize changing minds does not go against the natural grain of learning. Rather, it puts a nice finish on it to protect and accentuate the natural beauty of that process. Each suggestion above moves us in the right direction, and collectively they send a powerful message about what we value.

And that is just a few of the things we can do. I am proud to say that the Inspiring Connections courses (available now, ask your Regional Professional Learning Coordinator and/or attend the Inspiring Connections session at CPM’s National Teacher Conference!) include all this and much more. So, that’s what I’ve been thinking. What are your thoughts? How are you normalizing iteration, evolving ideas, and kaizen?

*Dakabibi™ refers to a puzzle with a set of numbers and several empty boxes that need to be filled while meeting certain conditions. For example, using the digits 0 through 9 at most once each, arrange the digits to create a true equation.

[ Responses may vary. For example, _ 1/4 = 0 .25, _ 3/4 = 0.75, or _6/5 = 1.20. ]

The term Dakabibi comes from the Twi phrase “adaka a bibiara ?ni mu,” which translates literally to “a box that everything is not inside.” This phrase is shortened to Dakabibi in CPM courses. Twi is a language spoken in Ghana and is one of the more widely used of over 50 languages spoken there. Like other West African languages, Twi is a tonal language; it has several phonemes that are very difficult for non-native speakers to pronounce. Almost a third of the Ghanaian population speaks Twi as a first or second language. Twi is not the most-spoken language in Ghana, but it is the most-spoken language by Ghanaians in America.

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Algebra Tiles Blue Icon

Algebra Tiles Session

  • Used throughout CPM middle and high school courses
  • Concrete, geometric representation of algebraic concepts.
  • Two-hour virtual session,
  •  Learn how students build their conceptual understanding of simplifying algebraic expressions
  • Solving equations using these tools.  
  • Determining perimeter,
  • Combining like terms,
  • Comparing expressions,
  • Solving equations
  • Use an area model to multiply polynomials,
  • Factor quadratics and other polynomials, and
  • Complete the square.
  • Support the transition from a concrete (manipulative) representation to an abstract model of mathematics..

Foundations for Implementation

This professional learning is designed for teachers as they begin their implementation of CPM. This series contains multiple components and is grounded in multiple active experiences delivered over the first year. This learning experience will encourage teachers to adjust their instructional practices, expand their content knowledge, and challenge their beliefs about teaching and learning. Teachers and leaders will gain first-hand experience with CPM with emphasis on what they will be teaching. Throughout this series educators will experience the mathematics, consider instructional practices, and learn about the classroom environment necessary for a successful implementation of CPM curriculum resources.

Page 2 of the Professional Learning Progression (PDF) describes all of the components of this learning event and the additional support available. Teachers new to a course, but have previously attended Foundations for Implementation, can choose to engage in the course Content Modules in the Professional Learning Portal rather than attending the entire series of learning events again.

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Building on Instructional Practice Series

The Building on Instructional Practice Series consists of three different events – Building on Discourse, Building on Assessment, Building on Equity – that are designed for teachers with a minimum of one year of experience teaching with CPM instructional materials and who have completed the Foundations for Implementation Series.

Building on Equity

In Building on Equity, participants will learn how to include equitable practices in their classroom and support traditionally underserved students in becoming leaders of their own learning. Essential questions include: How do I shift dependent learners into independent learners? How does my own math identity and cultural background impact my classroom? The focus of day one is equitable classroom culture. Participants will reflect on how their math identity and mindsets impact student learning. They will begin working on a plan for Chapter 1 that creates an equitable classroom culture. The focus of day two and three is implementing equitable tasks. Participants will develop their use of the 5 Practices for Orchestrating Meaningful Mathematical Discussions and curate strategies for supporting all students in becoming leaders of their own learning. Participants will use an equity lens to reflect on and revise their Chapter 1 lesson plans.

Building on Assessment

In Building on Assessment, participants will apply assessment research and develop methods to provide feedback to students and inform equitable assessment decisions. On day one, participants will align assessment practices with learning progressions and the principle of mastery over time as well as write assessment items. During day two, participants will develop rubrics, explore alternate types of assessment, and plan for implementation that supports student ownership. On the third day, participants will develop strategies to monitor progress and provide evidence of proficiency with identified mathematics content and practices. Participants will develop assessment action plans that will encourage continued collaboration within their learning community.

Building on Discourse

In Building on Discourse, participants will improve their ability to facilitate meaningful mathematical discourse. This learning experience will encourage participants to adjust their instructional practices in the areas of sharing math authority, developing independent learners, and the creation of equitable classroom environments. Participants will plan for student learning by using teaching practices such as posing purposeful questioning, supporting productive struggle, and facilitating meaningful mathematical discourse. In doing so, participants learn to support students collaboratively engaged with rich tasks with all elements of the Effective Mathematics Teaching Practices incorporated through intentional and reflective planning.